A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ______ - Physics

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A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ______

Options

  • `1/"e"`

  • `("e" - 1)/"e"`

  • `"e"/("e" - 1)`

  • `"e"/2`

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Solution

A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is `underline(("e" - 1)/"e")`.

Explanation:

Mean life time `tau = 1/lambda`

Now, N = `"N"_0"e"^{-lambda"t"}`

∴ For t = τ = `1/lambda`

`"N"/"N"_0 = "e"^{-lambda xx 1/lambda} = 1/"e"`

∴ The fraction that has disintegrated is given as,

`1 - "N"/"N"_0 = 1 - 1/"e" = ("e" - 1)/"e"`

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